I'm not convinced yet. The free algebra on a vector space is a left adjoint to the forgetful functor {Algebras}→{Vector Spaces}. Composing it with a right adjoint to {Hopf Algebras}→{Algebras} wouldn't give you an adjoint to the forgetful functor {Hopf Algebras}→{Vector Spaces}.
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Anton GeraschenkoApr 27 '10 at 1:03

I agree that this is a cute piece of information ... I was vaguely wondering if there's an adjoint on the other side too, and now I know!
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Anton GeraschenkoApr 27 '10 at 1:39

Here's that answer in a slightly more high-falutin' language. Let's have a category, $\mathcal{D}$. Then for any Lawvere theory, say $\mathcal{V}$, we can consider $\mathcal{V}$-algebra objects in $\mathcal{D}$; let's call this $\mathcal{D}\mathcal{V}$. If $\mathcal{D}$ has some nice structure, then the forgetful functor $\mathcal{D}\mathcal{V} \to \mathcal{D}$ has a left adjoint, the free $\mathcal{V}$-algebra functor.

But we can play this game in the opposite category as well. Only instead of simply going through the mirror and staying there, we go through, do our construction, and then come back again. So we form the category of co-$\mathcal{V}$-algebra objects in $\mathcal{D}$ by taking the opposite category of the category of $\mathcal{V}$-algebra objects in the opposite category of $\mathcal{D}$. Confused? Here it is in notation:

The double-opping means that the forgetful functor from $\mathcal{D}\mathcal{V}^c$ goes to $\mathcal{D}$ and not $\mathcal{D}^{op}$. However, as we've just gone through a mirror and back again, if our category $\mathcal{D}$ has some co-nice structure, $\mathcal{D}\mathcal{V}^c \to \mathcal{D}$ has a right adjoint, as requested.

In the cases you ask about, the category $\mathcal{D}$ is the opposite category of the category of models in either Set or $\operatorname{Set}_*$ (pointed sets) of some Lawvere theory. In each case, the Lawvere theory is based on algebras and so we tend to write its coproduct as the tensor product (though, of course, it's a slightly different tensor product in each case). The category of Hopf-algebras is then formed by taking an appropriate Lawvere theory, $\mathcal{V}$: possibly monoids, possibly groups, and forming $\mathcal{D}\mathcal{V}$ and then taking the opposite category again.

So the question boils down to whether or not the category of models of a Lawvere theory in either $Set$ of $\operatorname{Set}_*$ has enough co-nice structure and whether or not what we traditionally think of as Hopf algebras truly is the category of co-whatsits in whatevers (or whether, as in the case of Banach spaces, there are a few extra things floating around that we didn't think of). Fortunately, these are both true.